\documentclass{gatech-thesis}
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\begin{document}

\newcommand{\ds}{\displaystyle}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{proposition}[theorem]{Proposition}
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{corollary}[theorem]{Corollary}
\newtheorem{definition}[theorem]{Definition}
\newtheorem{example}[theorem]{Example}
\newtheorem{remark}[theorem]{Remark}

\section{Vector Algebra}

\begin{definition}
	$\vec V \in \ds \mathbb{R}^2$ is a quantity of both magnitude $\ds \big |\vec V\big |$ and direction. 
\end{definition}
\begin{definition}
	$\vec V = V_x\hat{i} + V_y\hat{j}$ where $\hat{i}$ and $\hat{j}$ are the unit vectors in the Cartesian $xy$ context.	
\end{definition}


\begin{theorem}[Magnitude of a Vector]
	$\big |\vec V\big | = \ds \sqrt{\ds \big(V_x\big)^2 + \big(V_y\big)^2}$.
\end{theorem}

\begin{theorem}[Direction of a Vector]
	
	The direction of $\vec V$ can be found by calculating the angle $\alpha$ with respect to the x-axis using $\tan(\alpha) = \ds \frac{V_y}{V_x}$. 
\end{theorem}

\begin{remark} The resulting $\alpha$ is in radians, and you may want to convert it to degrees by multiplying by $\ds \frac{180}{\pi}$.
\end{remark}

\begin{remark} In some instances, inverse trigonometric functions are used to find the angle.	
\end{remark}

\normalfont

\begin{theorem}[Summing of Vectors]
	Given $\ds \vec V = \ds V_x\hat{i} + V_y\hat{j}$ and $\vec W = W_x\hat{i} + W_y\hat{j}$, we sum their components separately to obtain the resultant vector $\vec R = \vec V + \vec W = (V_x + W_x)\hat{i} + (V_y + W_y)\hat{j}$.
\end{theorem}

\begin{theorem}[Difference of Vectors] Conversely, we subtract their components separately: $\vec V - \vec W = (V_x - W_x)\hat{i} + (V_y - W_y)\hat{j}$.
\end{theorem}

\subsection{Magnitude and Direction of Resultant Vectors}

For some sum or difference of two vectors $\vec R$, the above formulae may be used to find the magnitude and direction of $\vec R$.
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